Similarly, based on the definition of the Kronecker product \([{\mathbf {C}}^{(l,k)}_{i,y} \otimes {\mathbf {C}}^{(l,k)}_{i,z} ]_{M_{i,z} (r-1) +m \;\; M_{i,z} (s-1) +n } = [{\mathbf {C}}^{(l,k)}_{i,y} ]_{rs} [{\mathbf {C}}^{(l,k)}_{i,z} ]_{mn}\) , we can extract \({\mathbf {C}}^{(l,k)}_{i,y}\)  from \({\mathbf {C}}_i\)  corresponding to \([{\mathbf {C}}^{(l,k)}_{i,z} ]_{m m} = e^{j\pi (m-1) [\cos (\phi _{i,l}) - \cos (\phi _{i,k})] } \) . By doing this, the autocorrelation matrix \({\mathbf {C}}_{i,y}^{(m)}\)  for \(\sum _{l = 1}^{L_i}e^{j\pi (m-1) \cos (\phi _{i,l})}\gamma _{i,l}\alpha _y(\theta _{i,l},\phi _{i,l})\)  is obtained, which is the linear combination of the array response vectors \(\alpha _y(\theta _{i,l},\phi _{i,l})\)  for \(l = 1,2,\ldots ,L_i\) . Similarly, after getting \({\mathbf {C}}_{i,y} = \sum _{m=1}^{M_{i,z}}{\mathbf {C}}_{i,y}^{(m)}\) , we can extract the estimation of \(\theta _{i,1}\)  using via the root MUSIC algorithm [1]}, which, as previously mentioned, has computational complexity \(O\big (M_{i,y}^{3}\big )\) .
